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24 Bernoulli and Euler PolynomialsProperties

§24.10 Arithmetic Properties

Contents

§24.10(i) Von Staudt–Clausen Theorem

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Notes:
See Ireland and Rosen (1990, pp. 233–237) or Washington (1997, p. 56). For (24.10.2) see Carlitz (1953, p. 167).
Keywords:
Bernoulli numbers, congruence of rational numbers, von Staudt–Clausen theorem
Permalink:
http://dlmf.nist.gov/24.10.i

Here and elsewhere in §24.10 the symbol p denotes a prime number.

24.10.1\mathop{B_{{2n}}\/}\nolimits+\sum_{{(p-1)\divides 2n}}\frac{1}{p}=\hbox{% integer},

where the summation is over all p such that p-1 divides 2n. The denominator of \mathop{B_{{2n}}\/}\nolimits is the product of all these primes p.

24.10.2p\mathop{B_{{2n}}\/}\nolimits\equiv p-1\;\;(\mathop{{\rm mod}}p^{{\ell+1}}),

where n\geq 2, and \ell(\geq 1) is an arbitrary integer such that (p-1)p^{\ell}\divides 2n. Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference.

§24.10(ii) Kummer Congruences

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Notes:
See Ireland and Rosen (1990, pp. 239–241) or Washington (1997, p. 61). For (24.10.5) and (24.10.6) see Ernvall (1979, pp. 36 and 24). For historical remarks see Slavutskiĭ (1995) or Slavutskiĭ (1999).
Keywords:
Bernoulli numbers, Euler numbers, Kummer congruences
Permalink:
http://dlmf.nist.gov/24.10.ii

where m\equiv n\not\equiv 0\;\;(\mathop{{\rm mod}}p-1).

valid when m\equiv n\;\;(\mathop{{\rm mod}}(p-1)p^{\ell}) and n\not\equiv 0\;\;(\mathop{{\rm mod}}p-1), where \ell(\geq 0) is a fixed integer.

24.10.5\mathop{E_{{n}}\/}\nolimits\equiv\mathop{E_{{n+p-1}}\/}\nolimits\;\;(\mathop{{% \rm mod}}p),
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Symbols:
\mathop{E_{{n}}\/}\nolimits\!\left(x\right): Euler polynomials, n: integer and p: prime
Referenced by:
§24.10(ii)
Permalink:
http://dlmf.nist.gov/24.10.E5
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where p(>2) is a prime and n\geq 2.

24.10.6\mathop{E_{{2n}}\/}\nolimits\equiv\mathop{E_{{2n+w}}\/}\nolimits\;\;(\mathop{{% \rm mod}}2^{\ell}),
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Symbols:
\mathop{E_{{n}}\/}\nolimits\!\left(x\right): Euler polynomials, \ell: integer and n: integer
Referenced by:
§24.10(ii)
Permalink:
http://dlmf.nist.gov/24.10.E6
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valid for fixed integers \ell(\geq 0), and for all n(\geq 0) and w(\geq 0) such that 2^{\ell}\divides w.

§24.10(iii) Voronoi’s Congruence

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Notes:
See Uspensky and Heaslet (1939, p. 261) or Ireland and Rosen (1990, p. 237).
Keywords:
Bernoulli numbers, Voronoi’s congruence
Referenced by:
§24.10(iv), §24.19(ii)
Permalink:
http://dlmf.nist.gov/24.10.iii

Let \mathop{B_{{2n}}\/}\nolimits=\ifrac{N_{{2n}}}{D_{{2n}}}, with N_{{2n}} and D_{{2n}} relatively prime and D_{{2n}}>0. Then

24.10.7(b^{{2n}}-1)N_{{2n}}\equiv{2nb^{{2n-1}}D_{{2n}}\sum_{{k=1}}^{{M-1}}k^{{2n-1}}% \left\lfloor\frac{kb}{M}\right\rfloor\;\;(\mathop{{\rm mod}}M)},

where M(\geq 2) and b are integers, with b relatively prime to M.

For historical notes, generalizations, and applications, see Porubský (1998).

§24.10(iv) Factors

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Notes:
For (24.10.8) see Ribenboim (1979, p. 105) or Girstmair (1990a). For (24.10.9) see Carlitz (1954a).
Keywords:
Bernoulli numbers, Euler numbers
Permalink:
http://dlmf.nist.gov/24.10.iv

With N_{{2n}} as in §24.10(iii)

24.10.8N_{{2n}}\equiv 0\;\;(\mathop{{\rm mod}}p^{\ell}),
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Symbols:
\ell: integer, n: integer and p: prime
Referenced by:
§24.10(iv)
Permalink:
http://dlmf.nist.gov/24.10.E8
Encodings:
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valid for fixed integers \ell(\geq 1), and for all n(\geq 1) such that 2n\not\equiv 0 \;\;(\mathop{{\rm mod}}p-1) and p^{\ell}\divides 2n.

24.10.9\mathop{E_{{2n}}\/}\nolimits\equiv\begin{cases}0\;\;(\mathop{{\rm mod}}p^{\ell% })&\text{if }p\equiv 1\;\;(\mathop{{\rm mod}}4),\\ 2\;\;(\mathop{{\rm mod}}p^{\ell})&\text{if }p\equiv 3\;\;(\mathop{{\rm mod}}4)% ,\end{cases}
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Symbols:
\mathop{E_{{n}}\/}\nolimits\!\left(x\right): Euler polynomials, \ell: integer, n: integer and p: prime
Referenced by:
§24.10(iv)
Permalink:
http://dlmf.nist.gov/24.10.E9
Encodings:
TeX, pMML, png

valid for fixed integers \ell(\geq 1) and for all n(\geq 1) such that (p-1)p^{{\ell-1}}\divides 2n.

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